Description: Obsolete version of sbalex as of 14-Aug-2025. (Contributed by NM, 14-Apr-2008) (Revised by BJ, 20-Dec-2020) (Revised by BJ, 21-Sep-2024) (Proof modification is discouraged.) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | sbalexOLD | ⊢ ( ∃ 𝑥 ( 𝑥 = 𝑡 ∧ 𝜑 ) ↔ ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nfa1 | ⊢ Ⅎ 𝑥 ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) | |
| 2 | ax12v2 | ⊢ ( 𝑥 = 𝑡 → ( 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ) ) | |
| 3 | 2 | imp | ⊢ ( ( 𝑥 = 𝑡 ∧ 𝜑 ) → ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ) |
| 4 | 1 3 | exlimi | ⊢ ( ∃ 𝑥 ( 𝑥 = 𝑡 ∧ 𝜑 ) → ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ) |
| 5 | equs4v | ⊢ ( ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) → ∃ 𝑥 ( 𝑥 = 𝑡 ∧ 𝜑 ) ) | |
| 6 | 4 5 | impbii | ⊢ ( ∃ 𝑥 ( 𝑥 = 𝑡 ∧ 𝜑 ) ↔ ∀ 𝑥 ( 𝑥 = 𝑡 → 𝜑 ) ) |